This is not a real savings plan if you are robbing Peter to pay for Paul.
You take $100 cash out of the bank to buy groceries for the week, per your budget. You stop at the grocery store for a few items and pay cash. In your change you find a $5, so you put in into a jar. You do this three more times before the week is out. At the end of the week you have $20 in the jar but find out you are out of money for groceries and do not have enough food to last until the next $100 draw.
Off to the bank you go and you take out just $20 to buy milk, bread, eggs, fruit and veggies. This takes all the $20.
At the end of the week you spent $100 on groceries and put $20 in your jar.
You took out $120 from the bank.
You did not save any money but just moved money from the bank to a jar. This is funny money accounting. It might make you feel better thinking you have extra money to spend but you do not. You are now overspending because of that "found" money.
The best way to save it by a budget. Each pay period you put $X into long-term savings. $Y into short-term savings. $Z into the vacation/movie/fun money savings.